Euclidean spaces are nowadays defined in terms of Cartesian coordinates,
but they have various definitions.

E as a topological space

The one-dimensional Euclidean space E is defined on the set of
real numbers taken in their natural order.

Given some real numbers r, e, consider the set {n| r-e < n
< r+e}.

Let us construct the collection S of
all such subsets for any r and any e. The Euclidean topology is the topology
for which S is a base. (That is, any open interval or
any combination of open intervals is considered "open" in this topology).

The N-dimensional Euclidean space EN
is the product space of N copies of E. For example, a set
in E2 is open if it is the union of open rectangles,
and a set in E3 is open if it is the union of open rectanglular
prisms.